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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 109200.ge
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 109200.ge1 | 109200fz8 | \([0, 1, 0, -16076182008, 784547998715988]\) | \(7179471593960193209684686321/49441793310\) | \(3164274771840000000\) | \([2]\) | \(63700992\) | \(4.0856\) | |
| 109200.ge2 | 109200fz6 | \([0, 1, 0, -1004762008, 12258295075988]\) | \(1752803993935029634719121/4599740941532100\) | \(294383420258054400000000\) | \([2, 2]\) | \(31850496\) | \(3.7391\) | |
| 109200.ge3 | 109200fz7 | \([0, 1, 0, -992414008, 12574280395988]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-5753695101847907760000000000\) | \([2]\) | \(63700992\) | \(4.0856\) | |
| 109200.ge4 | 109200fz5 | \([0, 1, 0, -198562008, 1075106675988]\) | \(13527956825588849127121/25701087819771000\) | \(1644869620465344000000000\) | \([2]\) | \(21233664\) | \(3.5363\) | |
| 109200.ge5 | 109200fz3 | \([0, 1, 0, -63570008, 186566483988]\) | \(443915739051786565201/21894701746029840\) | \(1401260911745909760000000\) | \([2]\) | \(15925248\) | \(3.3925\) | |
| 109200.ge6 | 109200fz2 | \([0, 1, 0, -16562008, 4582675988]\) | \(7850236389974007121/4400862921000000\) | \(281655226944000000000000\) | \([2, 2]\) | \(10616832\) | \(3.1898\) | |
| 109200.ge7 | 109200fz1 | \([0, 1, 0, -10290008, -12640236012]\) | \(1882742462388824401/11650189824000\) | \(745612148736000000000\) | \([2]\) | \(5308416\) | \(2.8432\) | \(\Gamma_0(N)\)-optimal |
| 109200.ge8 | 109200fz4 | \([0, 1, 0, 65085992, 36425395988]\) | \(476437916651992691759/284661685546875000\) | \(-18218347875000000000000000\) | \([2]\) | \(21233664\) | \(3.5363\) |
Rank
sage: E.rank()
The elliptic curves in class 109200.ge have rank \(0\).
Complex multiplication
The elliptic curves in class 109200.ge do not have complex multiplication.Modular form 109200.2.a.ge
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
